Spectrally additive group homomorphisms on Banach algebras

Let A and B be unital complex Banach algebras. A surjective map ϕ:A→B is called a spectrally additive group homomorphism if the spectrum of x±y is equal to the spectrum of ϕ(x)±ϕ(y) for each x,y∈A. If A is semisimple and either A or B has an essential socle, then we prove that a spectrally additive group homomorphism ϕ:A→B is a continuous Jordan-isomorphism. If, in addition, either A or B is prime, then we conclude that ϕ is either a continuous algebra isomorphism or anti-isomorphism. It is noteworthy that the continuity and linearity (or even additivity) of the map ϕ does not form part of the hypothesis, but is rather obtained in the conclusion. The techniques employed in the proof of these results utilizes the spectral rank, trace and determinant, and yields a new additive characterization of finite rank elements in a Banach algebra which is of independent interest.